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Chapter 8: Q3 E (page 472)

Sketch the p.d.f. of the\({{\bf{\chi }}^{\bf{2}}}\)distribution withmdegrees of freedom for each of the following values ofm. Locate the mean, the median, and the mode on each sketch. (a)m=1;(b)m=2; (c)m=3; (d)m=4.

Short Answer

Expert verified
  1. Graph for the degree of freedom m=1
  2. Graph for the degree of freedom m=2
  3. Graph for the degree of freedom m=3
  4. Graph for the degree of freedom m=4

Step by step solution

01

Given information

Here the p.d.f of the chi-square distribution with m degree of freedom is given

02

The formula for mean, median, and mode

\(\begin{align}Mean &= m\\Median &= m{\left( {1 - \frac{2}{{9m}}} \right)^3}\\Mode &= \max \left\{ {m - 2,0} \right\}\end{align}\)

03

Step 3:: Graph of the chi-square distribution with 1 degree of freedom

\(\begin{align}Form &= 1\\Mean &= 1\\Median &= 0.4705075\\Mode &= 0\end{align}\)

04

Graph of the chi-square distribution with 2 degree of freedom

\(\begin{align}Form &= 2\\Mean &= 2\\Median &= 1.404664\\Mode &= 0\end{align}\)

05

Graph of the chi-square distribution with 3degree of freedom

\(\begin{align}Form &= 3\\Mean &= 3\\Median &= 2.381497\\Mode &= 1\end{align}\)

06

Graph of the chi-square distribution with 4 degree of freedom

\(\begin{align}Form &= 4\\Mean &= 4\\Median &= 3.369684\\Mode &= 2\end{align}\)

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Suppose that \({X_1},...,{X_n}\) form a random sample from the exponential distribution with unknown mean 渭. Describe a method for constructing a confidence interval for 渭 with a specified confidence coefficient \(\gamma \left( {0 < \gamma < 1} \right)\)

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